Polynomial upper bounds for the instability of the Nonlinear Schr\"odinger equation below the energy norm

We continue the study (initiated in \cite{ckstt:7}) of the orbital stability of the ground state cylinder for focussing non-linear Schr"odinger equations in the $H^s({\mathbb{R}}^n)$ norm for $1-\text{eps}<s<1$, for small $\text{eps}$. In the $L^2$-subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the $H^1$-subcritical case then we cannot show this, but for defocussing equations we obtain global well-posedness and polynomial growth of $H^s$ norms for $s$ sufficiently close to 1.